Random thoughts about teaching math

Concrete, Pictorial and Symbolic

In my post on the Alberta Math Dialogue, in which a group of Alberta university professors got together and offered their critique of our current curriculum, I mentioned that I heard some things that really offended me. Two of them aren’t even worth elaborating on (and for the record, were not uttered by a university math professor). Two of them had to do with math and I addressed one in a previous post, in which I discussed how I sometimes wonder if university math professors truly understand who we teach in K-12 math. It has some relevance here. It’s ridiculously long. Perhaps you should read it first. The second comment that offended me had to do with the use of concrete (hands on materials) and pictorial (drawing) representations.

In her critique of the Junior High curriculum, Christina Anton from Grant MacEwan University, talked about visiting a junior high math classroom and seeing the students colouring and using fabrics. From the context she described (polynomials), I suspect she saw a frugal teacher who had made algebra tiles out of old fabric, rather than spending sparse school money on a commercial set. Because she got a good laugh out of this, Christina kept coming back to it, and it became the running joke of the day. The Edmonton Journal even published the joke.

It may come as a surprise to you, as it did to me, but Grade 9 students here are required to use sticks, tiles, swatches of cloth and colouring to do complex math operations such as multiplying polynomials with monomials.

Here’s the thing, though. It’s not funny. After her session, I offered to show her how algebra tiles connect to base 10 blocks and make a nice bridge to symbolic algebra in grade 10. Christina dismissed me, and stated emphatically that concrete and pictorial representations are not real mathematics and have no place in the junior high curriculum. Only symbolic representations (the x’s and y’s and so on) are real mathematics and they are the only things that should be taught.

Such statements show the true naiveté of (some, not all) mathematics professors about who we teach in K-12 schools, and how those students learn. Concrete and pictorial representations help students make the jump to symbolic. For many students, they help form a critical bridge to understanding.

It is true that many of our students can make the jump to symbolic representations fairly quickly. But even those students still benefit from the bridge that concrete and pictorial representations make to that symbolic notation. We could probably even leave out the concrete and pictorial for our strongest students and they would be able to replicate the algebra without too much difficulty. The manipulatives will deepen their understanding, though.

For our visual and tactile learners, though, these concrete and pictorial representations are absolutely critical pieces. That’s no joke.

Would I force a student who can do it symbolically to draw it for me on an assignment or test? No. Would I let a student who can’t do it symbolically show me concretely or pictorially instead? Certainly. Would I expect a student bound for university calculus to be able to do it symbolically? Absolutely.

Do we still like what Singapore is doing? To those who speak derisively about concrete and pictorial representations, I leave you with the Singapore Bar Model. (Sorry, that was the best video I could find quickly with a google search.) The Singapore Bar Model creates lovely pictorial representations that help students make the bridge to symbolic notation. These representations work, even for high school algebra.

That particular homework sheet, by the way, comes from Ontario, in a district where there is strict use of their 4-point grading scheme in which students receive 4 only for “showing understanding”. In later exercises such as factoring trinomials or completing the square, how will your “students who make the jump to symbolic” fare in that system against students who have real trouble with the algebraic process but who can successfully juggle diagrams of squares and rectangles on paper to get the same answer in an ad-hoc fashion? And … when they consistently get 2 versus students completely unprepared for advancing in algebra, who consistently get 4, what is their teacher to infer from the results? Who needs remediation? Based on that numerical evaluation, who should be advanced into Applied or Precalculus versus the Consumer math pathway?

As a math teacher in Ontario, a student who demonstrates understanding using manipulatives does “show[ing] understanding” therefore deserving the highest achievement. As for completing the square, I have yet someone show me “the square” they completed algebraically. I can clearly see the square being completed using algebra tiles and in fact have students complete the square (even complex trinomials) without writing any algebra at all in a matter of seconds. So, I think these students fare much better than the purely “algebraic” students.

Are you quite familiar with the Singapore materials? Do you have copies of the Singapore Primary Math series?

I work with kids in Grades 4 – 6 regularly. I have two favourite programs: the first is JUMP Math and the second is Singapore Math. I have a slight preference for the second but, from what I’ve noticed, JUMP works better for most kids.

A couple of points:

1. Cristina’s talk was about the Grades 7 – 9 curriculum. This Singapore bar model appears in the Singapore resources early (possibly as early as Grade 2 or 3). By Grade 6, symbolic algebra is taught in these resources (and there’s plenty of drill). I have taught the Singapore bar model method to kids and I like it a lot. However, it is very important that students move past models and onto symbolic. Relying on models limits what they can do and, actually, some kids find models distracting and confusing.

2. You seem to agree with me on a couple of points. You wouldn’t force kids to work with models on an assignment or test. You seem to realize the importance of symbolic mathematics. From what I’ve noticed, there seem to be misunderstandings in schools and among some consultants about the degree to which manipulatives should be used. Base 10 blocks are often treated as actual methods for working out simple calculations. I think this is preposterous. Base 10 blocks should be used to explain place value, regrouping, etc, but when it comes to working on actual problems, students should be encouraged to work symbolically. Again, think Roman numerals. Our ancestors abandoned such techniques for doing calculations hundreds of years ago. It’s regressive to use models and concrete materials to work through actual calculations. I think some time should be spent with manipulatives in K-6 but I think more time should be spent on symbolic math.

3. If you really like the Singapore approach, perhaps you can influence your government to list the Singapore Primary Math series as a recommended resource for teaching K-6 math. These resources are far superior to Math Makes Sense or Math Focus. Remember, though, that in addition to getting the Singapore bar technique that’s featured in those resources, we’ll also have to take the drill, the symbolic, and the standard algorithms that are prominent features of those materials. We could go even further and adopt the Singapore curriculum, exactly as written. That would make all of us very happy. It is a fantastic curriculum.

Anna, I’m not as familiar with the Singapore curriculum as you are. I’ve seen the bar model in action on some systems of equations problems, but not at lower levels. I am interested in learning more. One thing Steve Leinwand told me last weekend is that the Singapore resources also include multiple strategies. He talked about examples asking students to compare two different solutions to the same problem. To your knowledge, is that kind of thing there?

There are not a lot of multiple strategies in those resources (at least not in the same way you see this in Math Makes Sense). There are worksheets on mental math (but not for single-digit times tables – they require times table memorization by Grade 3). You could order the Singapore Primary Math series for yourself. I order them from Heritage Resources: http://www.heritageresources.ca/cat_search.asp

There are two parts to the main program for each grade: Primary Math Series A and Primary Math series B. There is a teacher text and a student workbook for both A and B.

In addition to the regular primary math series, they have intensive practice books and books with challenging word problems (the word problems are really good).

Here are a couple of Singapore Math placement tests, which would tell you which book to place at student in.

(Note fraction arithmetic, percentages – concepts not covered until Grade 7/8 in our curriculum. These books follow the Singapore curriculum and those concepts are covered in Grades 4/5 in their curriculum. I have a serious problem with delaying fraction arithmetic until Grade 7/8, by the way.)

It is very unfortunate that our brief interaction is marked by so many miss-understandings. I remember that during our very short discussion you mentioned the analogy between using the powers of 10 and the powers of x and how nice this is done with sticks. I am not sure if I understood this right, but for me it meant the analogy between, for example 324=3*10^2+2*10+4 and the polynomial 3*x^2+2*x+4.
I have no idea how you represent this with sticks, but I think it is a very ingenious analogy and very useful for teaching addition of polynomials. I don’t mind the sticks or blocks- on the contrary. After 20 years of teaching I believe it is great to be able to use as many tools as possible (especially visual, concrete) when a new concept is introduced, and I mentioned it in my talk. I agree with you that these tactile, visual tools benefit all the students.

In my talk I spoke against what I see in the math curriculum as an overemphasizing of “concrete models with tiles, fabrics, etc, or pictorial representation involving coloring”.
Although I value them as intuitive, pedagogical tools, I don’t think that modeling mathematical concepts concretely and pictorially should be an achievement indicator in junior high school.

I totally agree with your approach on testing students’ understanding of symbolic, concrete and pictorial modelling, but I want to point out that this very reasonable approach is not the obvious one in the actual curriculum. Although the ability to work with the symbolic representation of polynomials is the ultimate goal (at least in my opinion), the curriculum treats the concrete, pictorial and symbolic representations in the same way. As Cornelia pointed out, it is this interpretation of the curriculum that results in long coloring assignments in junior high school for every student, including those who are ready to practice the symbolic work. Moreover, the junior high school teachers, whom I talked to, told me that one of the reasons the curriculum is so fragmented is because they have to spend so much time practicing with the students the concrete and pictorial modeling for each new concept. In my talk this over-fragmentation (3 years to teach the first order equation, etc) was mentioned as one of the issues that need to be revised.

Another miss-conception promoted by the actual curriculum is that in order to justify a mathematical result, we should be able to “prove it by making it” using concrete, pictorial modeling. I was told that in grade 8 the students are not allowed to use diagonal multiplication for proportions (i.e. if a/b=c/d then ad=cb) because they cannot justify it using these models. It can be formally justified very easily based on what they learned about multiplying with the same number on both sides of “=”.
In K-4, this miss-conception comes in a different flavour: in order to show understanding of addition or multiplication, a student must be able to do it using multiple methods, preferably using several personal strategies.

Finally, another issue related to concrete and pictorial modeling is the confusion regarding the so called “proofs by example” or using “guess methods”. I know that we can prove using a “counter-example”, but with an “example” we can at most illustrate a statement. I am not against using examples (on the contrary), or inductive reasoning. However, if we find an example that “works” it doesn’t mean that we found a “proof”. This should be better emphasized because the students are confused. For example on all first year calculus exams we have to write that “guessing or proofs by examples” are not valid proofs.

Here is an example from geometry. In grade 6 the students are taught that the sum of the angles of a triangle is 180. I was curious how this is justified before the parallel lines are introduced, and I found out from a teacher that it is done using sticks. Students make a triangle out of sticks, then they make separately each of its angles out of other sticks, they put the 3 angles together and notice that the sticks make a straight line, so the sum should be 180. This is a great visual, tactile illustration. It will help the students better remember what the sum is, and in fact it is based on the same idea as the classical formal proof. However, I think it should be followed by the remark that the statement was justified for a particular triangle. Can we be sure that it is true for all possible triangles? In grade 7, after the parallel lines are studied, this result can be revisited and formally justified, using alternating angles (which are out of the actual curriculum). So, again I see concrete modeling (and pictorial representations because in geometry drawing is essential) as a tool, while the ultimate goal should be deductive reasoning.

To conclude, I agree that concrete or pictorially models are very valuable tools in Mathematics, at any level. However, especially at junior high school level, they should be used as tools, not achievement indicators. In a revised form the curriculum should very clearly specify what to teach not how to teach. For junior high school, I think the symbolic representations in algebra and the deductive reasoning in geometry are the essential, age appropriate topics, and they should be promoted in the curriculum.

I don’t want to impose my ideas and I am very open to new ones. I believe that the revised Math curriculum should be based on the collaboration between teachers and mathematicians. I don’t think we have to be part of “different churches”. After all, teaching is an essential part of my professional life and Mathematics is a big part of yours.

I was at the math dialogue in Camrose, Dr Anton, and attended the K-12 sessions and round table. What I found difficult in the sessions was listening to the laughter in the crowd as, in particular, you and Dr Troitsky made comments about math teaching and curriculum, and how eager you both were to continue to (in my perception) “play to the crowd” in order to get more laughs. It was disheartening for me as an elementary teacher who sees the issue very differently.

When I was at the U of A in my Math Curriculum class, I heard about algebra tiles for the first time. And for the next few years, I absolutely hated them. To me, they were these things that were tiny and flew really well and had no purpose but to confuse and confound me.

That is, until I started using them to complete the square.

Oh my gosh, did it make a difference. My kids were not memorizing a series of meaningless steps (to be honest, *I* didn’t even understand why completing the square worked) and using algebra tiles to complete literal squares. The process made so much more sense, both in a concrete way and in symbolic way.

Do I “force” my students to use algebra tiles? No. Do I test my students on their use of algebra tiles? No. Did the use of algebra tiles help my students understand the process of completing the square? Yes.

Did I have to answer “Why do we divide by 2 and square it?” with “Because that’s the way it is”? No. Thank goodness.

I resisted them, too. Hated them. Went straight to the symbolic with my kids. Once I understood an area model, and then tried the tiles, I saw their power. I don’t spend weeks on them. Just a class or two at the start of polynomials. Factoring and multiplying happen simultaneously. It’s a beautiful thing.

Teaching math as intended in our Alberta curriculum is challenging, and one thing missing from this discussion is an acknowledgement that teachers need on-going support to do it well. It’s not hard to find examples of terrible worksheets, or awful problems, or stories of teachers who are expecting 4 different strategies on a test when one is all that’s intended. People tend to obsess over those and discount any success stories from teachers on the front line. People who don’t teach children have no conception about what the job entails, day in and day out, and teachers need ongoing professional development as well as day-to-day support from colleagues and administrators to learn to do it well.

Only the best mathematicians & scientists understand the physicality of operators and formulas. While I agree with the Grant M. prof that many things are abstract and can’t readily be represented visually (10 space?), but I certainly disagree that this implies operations can’t be described from a physical perspective.